GR Ray Tracing

0. Aim

The goal here is to predict the path of a photon near a non-spinning, uncharged black hole. Spacetime near such a black hole is described by what is known as the Schwarzschild metric in the theory of general relativity. Without any loss of generality one can assume the starting location of a photon to be on the x-axis. Let the distance of this starting location be $r_0$ from the center and let $\delta_0$ be the angle that this photon makes with the x-axis. This is shown in the figure below.

sch.png

1. Qualitative predictions

Even before quantitatively solving for the null geodesic in Schwarzschild metric, some of the properties of the geodesic can be qualitatively ascertained based on the starting parameters $r_0$ and $\delta_0$. For this we first define a function $B(r)$, such that

(1)
\begin{align} B(r)=\frac{r}{\sqrt{1-\frac{2M}{r} }} \end{align}

and another quantity $b$, called the impact parameter, such that

(2)
\begin{align} b = B(r_0)\;sin(\delta_0). \end{align}

Once we have calculated $b$, the theory of general relativity predicts that (see, e.g. pg. 674 of MTW for details; the conditions noted below are almost directly quoted from there):

0. A photon starting at $r_0<2M$ will never escape outside $r=2M$. Thus $r=2M$ defines the size of a Schwarzschild black hole.

1. A photon starting at $2M<r_0<3M$ will eventually escape to infinity, rather than be captured by the black hole (i.e. go inside $r=2M$) if and only if $\delta_0<\pi/2$ and

(3)
\begin{align} sin(\delta_0) < \frac{3\sqrt 3 M }{B(r_0)}. \end{align}

2. A photon starting exactly at $r_0=3M$ will (1) eventually fall into the black hole if $\delta_0 > \pi/2$, or (2) go in a circular orbit around the black hole if $\delta_0 = \pi/2$, or (3) eventually escape to infinity if $\delta_0 < \pi/2$.

3. A photon starting at $r_0>3M$ will eventually escape to infinity if and only if: (1) $\delta_0 < \pi/2$, or (2) $\delta_0>\pi/2$ and

(4)
\begin{align} sin(\delta_0) > \frac{3\sqrt 3 M }{B(r_0)}. \end{align}

2. Quantitative calculations to compute a photon trajectory

For the Schwarzschild metric the equation for the trajectory of a photon, i.e. the null geodesic, is given by

(5)
\begin{align} \frac{d^2 u}{d \phi^2} = 3u^2 - u \end{align}

where $u=M/r$. This equation is obtained by differentiating Eq. 25.57 of MTW w.r.t. $\phi$. Since the physics here scales with the mass of the black hole ($M$), we will do all calculations assuming $M=1$. If needed, converting distance to physical units can be done by multiplying values of $r$ by $GM/c^2$. In other words, the radial coordinates here in units of gravitational radii.

In order to solve Eq.(5) numerically using routines e.g. like those in GNU Scientific Library (GSL), it is better to define a new quantity $w=du/d\phi$, which allows us to rewrite Eq.(5) into two first order differential equations:

(6)
\begin{split} \frac{du}{d\phi} &= w \\ \frac{dw}{d\phi} &= 3u^2 - u \end{split}

One of the initial conditions to solve Eq.(6) is simply

(7)
\begin{align} u_0 = \frac{1}{r_0} \end{align}

Finding the other initial condition, viz. $w_0$, is a bit trickier. To find $w_0$ we first note that the radial component of the photon's velocity at the starting point is $v_r = c\;cos (\delta_0) = cos (\delta_0)$. This is because $c=1$ in the system of units that we have chosen. Similarly the cross-radial component of the photon's starting velocity is $v_\phi = c\;sin (\delta_0) = sin (\delta_0)$. However elementary mechanics tells us that

(8)
\begin{split} v_r &= \frac{dr}{dt} \\ v_\phi &= r \frac{d\phi}{dt} \end{split}

This allows us to recast $w=du/d\phi$ as follows

(9)
\begin{split} w &= \frac{du}{d\phi} \\ &= \frac{du}{dr} \frac{dr}{d\phi} \\ &= -\frac{1}{r^2} \frac{dr}{d\phi} \\ &= -\frac{1}{r^2} \frac{dr/dt}{d\phi/dt} \\ &= -\frac{1}{r^2} \frac{cos(\delta)}{d\phi/dt} \\ &= -\frac{1}{r^2} \frac{cos(\delta)}{v_\phi/r} \\ &= -\frac{1}{r^2} \frac{cos(\delta)}{sin(\delta)/r} \\ &= -\frac{1}{r} \frac{1}{tan(\delta)} \\ &= -\frac{u}{tan(\delta)} \end{split}

Therefore the initial condition for $w_0$ is given by

(10)
\begin{align} w_0 = -\frac{u_0}{tan(\delta_0)} \end{align}

and the set of two differential equations in Eq.(6) can be solved using the two initial conditions given in Eq.(7) and Eq.(10) to obtain the trajectory of the photon.

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